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Outline

• Hierarchy of Information Levels

• Final-size models

• Survival models

• Likelihood models

• Bayesian models

• Summary

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Basic Setting for Household Studies

• A community of households. May consider neighborhoods.

• Infectious forces.

– Community at large: zoonotic source, infectious visitors.

– Within-household transmission.

– Between-household transmission.

• Symptom diary, e.g., headache, sore throat, fever.

• Lab-confirmation:

– Viral culture for nasal/throat swabs, often triggered by symptom

onset.

– HI titers: baseline and the end of study.

• Intervention implemented, e.g., vaccine vs. placebo.

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Hierarchy of Information Levels

• Consecutive occurence of infections is a counting process, observed at

different information levels (Rhodes, Halloran and Longini, JRSS B,

1996)

– How many infections have occurred in (0, T]. Final value models

(Longini et al, 1982; Addy et al, 1991).

– Times at which infection or symptom onset occurs. Survival model

(Longini and Halloran, 1996).

– Who contacts whom and/or who infects whom. Discrete-time

likelihood models (Rampey et al, 1992; Yang et al, 2006).

∗ Sometimes difficult to obtain.

∗ Clustering pattern is the bottom line.

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Final Size Model

• Longini and Koopman (Biometrics, 1982)

– B: Probability of escaping infection from external source during

epidemic.

– Q: Probability of escaping infection from an infectious household

member during epidemic.

– mjk: probability that j out of k household members are infected.

∗ Household with a single person: m01 = B and m11 = 1 −B

∗ Household with two members:

· m02 = B2

· m12 = 2(1 −B)BQ

· m22 = 1 −m02 −m12 = 2(1 −B)(1 −Q)B + (1 −B)2

∗ In general, mjk =(kj

)mjjB

k−jQj(k−j) and mjj = 1 −∑l

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– Maximum likelihood estimation

∗ Likelihood: L(B,Q) =∏k,jm

ajkjk , where ajk is the frequency of

households corresponding to mjk.

∗ Score function:

∂ lnL

∂B=

∑

k,j

ajk

{ 1mjj

(∂mjj∂B

)+k − j

B

}.

∗ Fisher’s information:

−E(∂2 lnL∂B2

)=

∑

k,j

nkmjk

{ 1m2jj

(∂mjj∂B

)2−

1

m2jj

∂2mjj∂B2

+k − j

B2

}.

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∗ Rough estimates for starting point

a0knk

= m̂0k = B̂kk ⇒ B̂k =

(a0knk

)1/k⇒ B̂ =

1

n

∑

k

nkB̂k

a1knk

= m̂1k = k(1 − B̂)B̂k−1Q̂k−1

Q̂φ̂B̂ ≈ 1 − θ̂ =⇒ Q̂ ≈(1 − θ̂B̂

)1/φ̂

where φ̂ =P

k,j jajkn and θ̂ =

P

k,j(jajk/k)

n .

• Inter-group mixing (Addy, Longini and Haber, Biometrics, 1991).

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Frailty Hazard Model

• Longini and Halloran (Applied Stat, 1996)

– αv: proportion of full immunity in group v (1 = vaccine, 0 =

control).

∗ If α1 > α0, “all-or-none”effect.

– θ: reduction rate in susceptibility for the 1 − α1 of vaccinated

population, “leaky”effect.

– Frailty (random) hazard

∗ Pr(Zv = 0) = αv

∗ Zv|Zv > 0 ∼ fv(mean = 1, variance = δv)

∗ Hazard function: λv(t) = Zvθvcπp(t).

∗ Survival function: Sv(t) = EZv

[exp

{−Zv

∫ t0λv(τ)dτ

}]

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– V E = 1 −E[λ1(t)

]

E[λ0(t)

] = 1 − (1−α1)θcπp(t)(1−α0)cπp(t) = 1 −(1−α1)(1−α0)

θ.

– For grouped survival data with k intervals, p(t) =∏ki=1 p

I(ti−1≤t

InfectiveSusceptible

p: within-household pairwise daily transmission probability without treatment.

b: daily probability of infection by the community without treatment (CPI).

AVES = : Efficacy of the antiviral agent in reducing susceptibility.

AVEI = : Efficacy of the antiviral agent in reducing infectiousness.

pθφ

1 φ−

bθ

b

1 θ−

pθ

pCommunity

pφ

Household

Transmission Patterns and Parameters of Interest

Time of

Infection

Latent period

(Incubation period)

Infectious period

0.2

0.6

0.2

0.3

0.7

0.1

1.0

Onset time of

symptoms and

infectiousness

1 2 3 4 5 6

1 2 3 4 5 6

Days

Days

Natural Disease History of Influenza

( | )g t t%

( | ) :g t t%

( | )f t t%

The probability of symptom onset on day given infection on

day t .

Probability that the host is infective on day t given symptom

onset on day .

( | ) :f t t%

t

t%

1.0

t%

t%

1.0

1 -

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Likelihood Model for Symptomatic Infection

Yang, Longini & Halloran (Appl. Stat., 2006)

• Likelihood for a person-day

Probability of pairwise transmission per daily contact:

pji(t) =

θri(t)φrj(t)pf(t|t̃j), j ∈ Hi

θri(t)b, j = c.

Define Di = Hi ∪ c. Probability of escaping infection on day t:

ei(t) =∏j∈Di

(1 − pji(t)

)

Probability of escaping infection up to day t:

Qi(t) =∏tτ=1 ei(τ)

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• Likelihood contributed by a single individual

If subject i is known to be infected on day t, the probability is

Ui(t) = [1 − ei(t)] Qi(t− 1),

Generally only symptom onset is observable

Li =

Qi(T ), if individual i is not infected∑tit=ti

g(t̃i|t)Ui(t), otherwise

where ti = t̃i − lmax, ti = t̃i − lmin and T is the last observation day for

the epidemic.

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• Selection bias in case-ascertained design: only households with infected

members are followed.

– Conditioning on the disease history (infection and symptom) up to

the symptom onset day of the index case t̃di .

Lm

i =

8

<

:

Li, index case,Pt̃di

t=1

n

Ui(t) Pr(t̃i > t̃di |t)o

+ Qi(t̃di), otherwise.

– Use the conditional likelihood Lci = Li/Lmi for inference.

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• Right-censoring: real-time analysis

– No symptoms observed could mean either escape from infection or

incubation period.

– Calculate the marginal probability of observing no symptom onset

up to day T:

Lmi = Qi(T − lmin)+

T−lmin∑

t=T−lmax+1

{(1−ei(t))Qi(t−1)

}×Pr(t̃i > T |t)

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• Assessing goodness of fit

– The probability of symptom onset on day t for subject i is

πi(t) =

t−lmin∑

τ=t−lmax

{(1 − ei(τ)

) τ−1∏

s=t−lmax

ei(s)}g(t|τ).

– Choose 0 = c0 < c1 < . . . < cm = 1, then n̂k =∑ck−1

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• Simulation study

Population: a community composed of households of size two or larger

with 1000 people is generated based on the age distribution and

household sizes from the US Census 2000.

Table 1: Empirical distributions of the latent period and the infectious period

(Elveback et al., 1976)

Latent Period Infectious Period

(days) Com. Prob. (days) Cum. Prob.

1 0.2 3 0.3

2 0.8 4 0.7

3 1.0 5 0.9

6 1.0

Table 2: Comparison of MLEs by randomization schemes and householdfollow-up schemesParameter‡ Estimate MonteCarlo 95%CIcoverage

standarderrors (%)§§I§ H§ I§ H§ I§ H§

θProspective 0.70 0.71 0.083 0.25 95.3 93.8

Case-ascertained 0.70 0.71 0.083 0.26 96.1 94.3φ

Prospective 0.20 0.24 0.045 0.16 94.6 91.5Case-ascertained 0.20 0.24 0.044 0.15 95.3 91.3

‡ True efficacy-related parameters are set to θ = 0.70 and φ = 0.20.§ I, individual-level randomization; H, household-level randomization.

§§ The 95% CI is obtained as exp[log(λ̂) ± 1.96 × se{log(λ̂)}]; λ = θ, φ.

Table 3: Two randomized multi-center trials of Oseltamivir, an influenzaantiviral agent.

Trial I Trial II(Welliver et al. 2001) (Hayden et al. 2004)

Time of trial 1998-1999 2000-2001Households 372 277Population 1329 1110Treatment for illness None OseltamivirDuration of medication

Illness treatment N/A 5 daysProphylaxis 7 days 10 days

Follow up (symptom diary) 14 days 30 daysInfected/Exposed(index) 165/372 179/298Infected/Exposed(susceptible)

Control† 38/464 45/392Oseltamivir 4/493 14/420

Table 4: Maximum likelihood estimates by age (1-17 vs 18+) for pooledoseltamivir trials conducted in 1998-1999 and 2000-2001, North Americaand Europe.

With Assumptionψ = θφ Parameter MLE 95% C.I.

Yes bc† 0.0023 (0.0015, 0.0035)

ba 0.00055 (0.0003, 0.001)pcc 0.038 (0.023, 0.063)pca 0.012 (0.007, 0.021)pac 0.018 (0.008, 0.040)paa 0.022 (0.014, 0.034)AVES 0.85 (0.52, 0.95)AVEI 0.66 (-0.10, 0.89)AVET 0.95 (0.77, 0.99)

No AVES 0.93 (0.50, 0.99)AVEI 0.78 (-0.27, 0.96)AVET 0.87 (0.41, 0.97)

SARcc‡ 0.15 (0.074, 0.21)

SARca 0.049 (0.021, 0.075)SARac 0.071 (0.014, 0.13)SARaa 0.086 (0.047, 0.12)

†, ‡ Subscription c denotes child (1-17), a denotes adult(18+), and ca denotes child-to-adult transmission.

‡ SARvu is based on the average 4.1 days of infectiousperiod, i.e., SARvu = 1 − (1 − pvu)

4.1.

Table 5: Assessing goodness-of-fit of the likelihood model† for pooledoseltamivir trials conducted in 1998-1999 and 2000-2001, North Americaand Europe.

Risk Total Observed # of Predicted # ofLevel Person-days illness onsets illness onsets1 2084 0 02 1321 0 03 15878 8 94 1434 1 35 8165 19 226 933 3 37 935 5 48 1241 12 99 1084 17 1810 894 25 27

† With assumption of ψ = θφ.

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Likelihood Model with Data Augmentation

Yang, Longini and Halloran (Comp. Stat. & Data Analysis, 2007)

• Data to be augmented

– Pairwise transmission outcome Yji(t) (1:transmission, 0:escape).

– Yji(t) is defined only if Yji(τ) = 0 for all τ < t.

– Yji(t) is not observed when j is infectious and ti ≤ t ≤ ti.

– Yji(t) is independent of Yki(t) for the same day t.

– More convenient to work with Zji(t) = Yji(t)∏k∈Di,τ

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Symptom onset

Potential transmissions not observable

1

2

4

3

1 11

2 2 2

3 3 3

4 4 4

c c c

Exposure Outcome

(1: transmission, 0:escape)

Expected

Frequencyi j�

1 2�

1 c�

0

1

0

1

21 1 1 1Pr[ ( 1) 1| ( ) ]Z t I t− =% %

21 1 1 1Pr[ ( 1) 1 | ( ) ]Z t I t− =% %

1 1 1 1Pr[ ( 1) 1 | ( ) ]cZ t I t− =% %

1 1 1 1Pr[ ( 1) 1 | ( ) ]cZ t I t− =% %

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• The likelihood of the augmented data

Li(b, p, θ, φ|t̃j, Zji(t), Z̄ji(t), j ∈ Di, t ≤ T )

=T∏

t=1

{g(t̃i|t)

maxj∈DiZji(t)∏

j∈Di

(pji(t))Zji(t)

(1 − pji(t)

)Z̄ji(t)},

where maxj∈DiZji(t) indicates if Zji(t) = 1 for any j on day t. The

log-likelihood is

log(Li(b, p, θ, φ|t̃j, Zji(t), Z̄ji(t), j ∈ Di, t ≤ T ))

∝

T∑

t=1

∑

j∈Di

{Zji(t) log(pji(t)) + Z̄ji(t) log(1 − pji(t))

},

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• The E-M algorithm Define the events

– Si(t): i has symptom onset on day t.

– Ii(t): i is infected on day t.

– Iji(t): j infects i on day t.

whose probabilities are given by

Pr[Iji(t)] = Q̂i(t− 1)p̂ji(t)

Pr[Ii(t)] = Q̂i(t− 1){1 − êi(t)

},

Pr[Si(t̃i)

]=

t̄i∑

τ=ti

g(t̃i|τ) × Pr[Ii(τ)

],

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The conditional distributions of Zji(t) and Z̄ji(t) are

Pr(Zji(t) = 1|b, p, θ, φ, t̃i) =

Pr[Iji(t)

]

Pr[Si(t̃i)

] × g(t̃i|t), ti ≤ t < t̄i

0, otherwise

and

Pr(Z̄ji(t) = 1|b, p, θ, φ, t̃i)

=

g(t̃i|t)×{

Pr[Ii(t)

]−Pr

[Iji(t)

]}

Pr[Si(t̃i)

] + ∑t̄iτ=t+1g(t̃i|τ)×Pr

[Ii(τ)

]

Pr[Si(t̃i)

] , ti ≤ t < t̄i

1, t < ti

0, otherwise

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• Variance estimation

Let Z = {Zji(t), Z̄ji(t)}, t̃ = {t̃i}, and λ = {b, p, θ, φ}. Louis’ method

states that

∂2 log(L(λ|t̃))

∂λ2= E

Z|t̃,λ

{−∂2 log(L(λ|t̃,Z))

∂λ2

}

+ VARZ|t̃,λ

{−∂ log(L(λ|t̃,Z))

∂λ

}

Table 6: Two randomized multi-center trials of zanamivir, an influenza an-tiviral agent

Hayden et al., 2000 Monto et al., 2002

Time of trial Oct. 1998 - Apr. 1999 Jun. 2000 - Apr. 2001Households 336 484Population 1186 1770Index case randomization Yes NoDuration of medication

Index case 5 days N/AContact 10 days 10 days

Follow up (symptom diary) 14 days 14 days

Infected†/Symptomatic(index) 164/336 281/484

Infected†/Exposed(contacts)Control 52/435 76/626Zanamivir 17/415 27/660

Numbers may slightly differ from references due to different criteria of data inclusion for analysis.† Laboratory-confirmed infections with clinical symptoms

Table 7: Estimates of efficacies and transmission probabilities by age (1-17vs. 18+) for pooled zanamivir trials conducted in 1998-1999 and 2000-2001.

IRLS MLEParameter Point Estimate SD Point Estimate SD 95% CI

bc† 0.0024 0.00052 0.0028 0.00063 (0.0017, 0.0042)

ba 0.00086 0.00030 0.0010 0.00039 (0.00045, 0.0021)

p†cc 0.040 0.0074 0.040 0.0077 (0.027, 0.057)pca 0.028 0.0045 0.029 0.0048 (0.021, 0.040)pac 0.023 0.0071 0.020 0.0071 (0.009, 0.037)paa 0.040 0.011 0.032 0.011 (0.016, 0.058)AVES 0.68 0.086 0.75 0.072 (0.56, 0.86)AVEI 0.24 0.38 0.23 0.44 (-1.33, 0.75)AVET 0.81 0.094 (0.50, 0.93)

† Subscript c denotes child (1-17), a denotes adult(18+), and ca denotes child-to-adult transmission.

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A Bayesian Model for Symptomatic Infection

Cauchemez et al. (Stat in Med., 2004)

• Assumption: infectious period (vi, ψi) follows Gamma(µ, σ).

• Observation level P (Y |v, ψ) =∏i∈I 1{Zi − 3 < vi < Zi and vi < ψi}

• Transmission level

λi(t) = αi + ǫi∑

j∈I(t)

βjn

P (v, ψ|θ) =∏

i∈I

dµ,σ(ψi − vi)∏

i∈I−{1}

λi(vi)e−

R

viv1λi(t)dt

∏

j∈S

e−

R 15v1λj(t)dt

SARi→j(n) = 1 −

∫ ∞

0

exp(−ǫj

βint)dµ,σ(t)dt

• Prior level

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A Bayesian Model for Asymptomatic Infections

Yang, Halloran & Longini (Biostatistics, 2009)

• Observed data

– H households, with household h of size nh.

– Enrollment day (ascertainment of index cases) as day 1.

– Follow-up period: {1, 2, . . . , Th}}.

– ILI onset dates: {t̃hi : i = 1, . . . , nh, h = 1, . . . , H}.

– Laboratory test results yhi.

• Latent variable

– Infection dates: {t̂hi : i = 1, . . . , nh, h = 1, . . . , H}.

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• Modeling viral transmission

– Exposure to constant risk γ0 from community, starting from day

Th < 1.

– Exposure to time-varying baseline risk γ1f(t−t̂hj

∆ |a, b) from

infected household member j during the infectious period

t̂hj ≤ t ≤ t̂hj + ∆.

∗ ∆: duration of infectious period, a known constant.

∗ f(·|a, b): beta density with parameters a and b.

∗ γ1: average risk over the infectious period, because

γ1 =1∆

∫ t̂hi+∆t̂hi

γ1f(t−t̂hi

∆ |a, b)dt

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– Risk adjusted for covariates:

λh,j→i(t) =

8

<

:

γ1f(t−t̂hj

∆ |a, b) exp{β′Sxhi(t) + β

′Ixhj(t)}, j > 0 and j 6= i,

γ0 exp{β′Sxhi(t)}, j = 0.

– Total risk: λhi(t) =∑j 6=i λh,j→i(t).

– Daily probability of escaping influenza infection:

qhi(t) = exp{−

∫ tt−1

λhi(τ)dτ}

.

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0 1 2 3 4 5 6 7

02

46

8

(a)

log 1

0TC

ID/m

l

0.0 0.2 0.4 0.6 0.8 1.0

0.0

1.0

2.0

(b)

Rel

ativ

e In

fect

ivity

log Viral Loadf(2.08, 2.31)f(4.68, 4.90)

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• Modeling pathogenicity (probability of developing ILI)

– Traditional assumption: the probability of ILI given infection,

ξhi(t), depends only on antiviral treatment status rhi(t) of

susceptible i on day t.

∗ ξhi(t) = η1−rhi(t)0 η

rhi(t)1 , where ηu is the probability of

developing ILI for rhi(t) = u, u = 0, 1.

∗ Traditional antiviral efficacy in reducing pathogenicity is

defined as AVEP = 1 − η1/η0.

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– Our assumption: ξhi(t) depends on antiviral treatment status of

the susceptible and all infectives in the household.

∗ Consider a single infective j.

· logit(ξhi(t)

)= α00 + α10rhj(t) + α01rhi(t).

· Let ηuv be the value of ξhi(t) for rhj(t) = u and rhi(t) = v,

then, ηuv = logit−1(α00 + vα01 + uα10)

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∗ For multiple infectious sources, consider the average

treatment status weighted by cumulative risks.

Λh,j→i(t) =

∫ t

t−1

λh,j→i(τ)dτ,

Λh,i(t) =

∫ t

t−1

λh,i(τ)dτ,

r̄h(t) =

nh∑

j=0

{Λh,j→i(t)

Λh,i(t)rhj(t)

},

logit(ζhi(t)

)= α00 + α10r̄h(t) + α01rhi(t).

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∗ Antiviral efficacies in reducing pathogenicity:

AVESp = 1 −η01η00

,

AVEIp = 1 −η10η00

,

AVETp = 1 −η11η00

.

∗ Antiviral efficacies for symptomatic infection:

1 − AVESd = (1 − AVESi)(1 − AVESp),

1 − AVEId = (1 − AVEIi)(1 − AVEIp),

1 − AVETd = (1 − AVETi)(1 − AVETp).

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• For the incubation period t̃hi − t̂hi, assume a known distribution

Pr(t̃hi|t̂hi).

• Joint probability of viral transmission and ILI:

– Define ω = (γ0, γ1,βS ,βI , α00, α01, α10, a, b).

Lhi

“

t̂hi, t̃hi | ω, {t̂hj : j 6= i}”

=

8

>

>

>

<

>

>

>

:

QTht=Th

qhi(t), t̂hi > Th,n

Qt̂hi−1

t=Thqhi(t)

on

1 − qhi(t̂hi)o

ξhi(t̂hi) Pr(t̃hi|t̂hi), Th ≤ t̂hi ≤ Th, t̃hi < ∞,n

Qt̂hi−1

t=Thqhi(t)

on

1 − qhi(t̂hi)o

`

1 − ξhi(t̂hi)´

, Th ≤ t̂hi ≤ Th, t̃hi = ∞,

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• Adjustment for selection bias

– Each household in the analysis has at least one infection.

– All index cases are symptomatic.

– Without adjustment, estimates of γ0, α00, α01 and α10 will be

biased.

– Adjustment: drop likelihood history up to (include) T̃h, the

symptom onset day of the index case.

Lhi

“

t̂hi, t̃hi | ω, {t̂hj : j 6= i}”

=

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

QTh

t=gTh+1qhi(t), t̂hi > Th,

n

Qt̂hi−1

t=gTh+1qhi(t)

on

1 − qhi(t̂hi)o

ξhi(t̂hi) Pr(t̃hi|t̂hi), fTh < t̂hi ≤ Th, t̃hi < ∞,n

Qt̂hi−1

t=gTh+1qhi(t)

on

1 − qhi(t̂hi)o

`

1 − ξhi(t̂hi)´

, fTh < t̂hi ≤ Th, t̃hi = ∞,

Pr(t̃hi|t̂hi), t̂hi ≤ fTh, t̃hi < ∞,

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• Full probability

– π(ω): the joint prior distribution.

– t̂ = {t̂hi : i = 1, . . . , nh, h = 1, . . . , H}.

– t̃ = {t̃hi : i = 1, . . . , nh, h = 1, . . . , H}.

– y = {yhi : i = 1, . . . , nh, h = 1, . . . , H}

– C(yhi|t̂hi): indicate whether yhi is compatible with t̂hi (1:yes,

0:no)

Pr(t̂,ω|y, t̃) ∝ Pr(y, t̂, t̃,ω)

=π(ω) ×

H∏

h=1

nh∏

i=1

Lhi

(t̂hi, t̃hi; | ω, {t̂hj : j 6= i}

)C(yhi|t̂hi).

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• MCMC sampling

– For parameters to be estimated, use random-walk style

Metropolis-Hastings’ algorithm.

– For unobserved infection times,

∗ the set of candidate infection days:

Ωhi =

8

<

:

{t : C(yhi|t) × Lhi(t, t̃hi|·) Pr(t̃hi|t) > 0}, symptomatic,

{t : C(yhi|t) × Lhi(t, t̃hi|·) > 0}, asymptomatic

∗ Sample t̂hi from

Pr(t̂hi = t|t̂hi ∈ Ωhi, ·)

=Lhi

“

t, t̃hi | ω, {t̂hj : j 6= i}”

Q

j 6=i Lhi

“

t̂hj , t̃hj | ω, {t̂hk : k 6= j}”

P

s∈ΩhiLhi

“

s, t̃hi | ω, {t̂hj : j 6= i}”

Q

j 6=i Lhj

“

t̂hj , t̃hj | ω, {t̂hk : k 6= j}” .

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• Data Analysis

– λh,j→i(t) = θRxRXiφRx

RXjθAgeAGEiφAge

AGEjγ1f(t−t̂hjδ |a, b).

– Identification of clinical symptom onset

∗ I: ≥ 37.8◦C plus cough or nasal congestion.

∗ II: ≥ 37.2◦C plus

any of (cough, nasal congestion, sore throat) and

any of (headache, aches/pains, chills/sweats, fatigue).

– Identification of candidate infection days

∗ A positive swab on day t indicates t− δ ≤ t̂hi ≤ t− 1.

∗ 4-fold increase in HI titers indicates 1 ≤ t̂hi ≤ Th given that

the subject is susceptible at baseline.

Table 8: Comparison between Bayesian estimates and previous findings.Parameter Bayesian Halloran et al. (2007)† Yang et al. (2006)‡

γ0 0.00046 (0.00006,0.0017) 0.00055 (0.0003,0.001)γ1 0.019 (0.0096,0.037) 0.022 (0.014,0.034)η00 0.50 (0.33,0.67) II: 0.57 (0.44,0.69)η01 0.29 (0.097,0.57) II: 0.12 (0.05,0.28)η10 0.082 (0.017,0.22)

AVESi 0.62 (0.39,0.77)

“ I : 0.48 (0.17,0.67)II : 0.64 (0.36,0.80)

”

AVEIi −0.18 (−0.93,0.30) 0.16 (−0.33,0.46)AVESp 0.41 (−0.28,0.81) II: 0.79 (0.45,0.92)AVEIp 0.84 (0.53,0.97)

AVESd 0.77 (0.45,0.93)

“ I : 0.81 (0.35,0.94)II : 0.91 (0.64,0.98)

”

0.85 (0.52,0.95)

AVEId 0.81 (0.42,0.96) 0.81 (0.45,0.93) 0.66 (−0.10,0.89)θAge 1.06 (0.64,1.91)φAge 1.05 (0.64,1.71)a 4.68 (1.44,16.86)b 4.90 (1.92,15.57)d 3.88 (3.03,4.48)

Table 9: Bayesian estimates by different infectiousness of asymptomaticcases relative to symptomatic cases.

Relative Infectiousness of Asymptomatic Infection1.0 0.5 0.3 0.2 0.1

γ0 0.00046 (0.00006,0.0017) 0.00063 (0.00009,0.0023) 0.0012 0.0025 0.0062 (0.0031,0.011)γ1 0.021 (0.011,0.038) 0.037 (0.019,0.067) 0.051 0.054 0.013 (0.0003,0.093)η00 0.49 (0.33,0.66) 0.48 (0.32,0.65) 0.45 0.38 0.29 (0.19,0.43)η01 0.30 (0.095,0.58) 0.31 (0.10,0.61) 0.32 0.34 0.35 (0.089,0.80)η10 0.080 (0.018,0.22) 0.077 (0.015,0.22) 0.076 0.079 0.063 (0.0,1.0)AVESi 0.61 (0.36,0.78) 0.61 (0.35,0.77) 0.62 0.63 0.54 (−0.14,0.86)AVEIi −0.21 (−0.94,0.29) −0.22 (−1.0,0.31) -0.29 -0.29 0.28 (−12.62,0.98)AVESp 0.38 (−0.32,0.81) 0.35 (−0.46,0.79) 0.26 0.099 −0.18 (−2.06,0.71)AVEIp 0.84 (0.53,0.96) 0.83 (0.49,0.97) 0.83 0.80 0.79 (−3.57,1.0)AVESd 0.77 (0.42,0.93) 0.75 (0.38,0.92) 0.72 0.68 0.50 (−0.46,0.86)AVEId 0.80 (0.38,0.96) 0.80 (0.32,0.96) 0.78 0.73 0.81 (−5.13,1.0)θAge 1.07 (0.64,1.87) 1.04 (0.64,1.74) 1.03 1.02 1.0 (0.59,1.77)φAge 1.05 (0.64,1.69) 0.91 (0.56,1.55) 0.81 0.76 1.85 (0.25,38.50)

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Figure 1: Sensitivity of the posterior median to the prior distribution foreach parameter, with flat priors for parameters other than the focal one.

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Discussion

• Methods designed for household studies may be generalizable to

other cluster settings.

• Simpler methods may be more robust to model mis-specification,

but may miss important information as well.

• Combining studies for meta analysis should be done carefully.

• Improve statistical inference via improving study design.

– Maximize information for targeted efficacy measures.

– individual level of randomization, including index cases.

– Complete symptom diary.

– Lab-tests at a higher frequency (e.g., the Hong Kong pilot NPI

study).

• Post-randomization bias.